On additive shifts of multiplicative subgroups

Vyugin, Il'ya Vladimirovich; Shkredov, Ilya D.

doi:10.1070/sm2012v203n06abeh004245

Cited by 22 publications

(24 citation statements)

References 19 publications

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“…In the special case of A being a multiplicative subgroup of F * p , the same bound was proved by Heath-Brown and Konyagin [17] and improved by V'jugin and Shkredov [40] (for suitably small multiplicative subgroups) to Ω . Theorem 3 becomes a vehicle to extend bounds for multiplicative subgroups to approximate subgroups.…”

Section: Resultssupporting

confidence: 60%

On the Number of Incidences Between Points and Planes in Three Dimensions

2017

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Abstract. We prove an incidence theorem for points and planes in the projective space P 3 over any field F, whose characteristic p = 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.This yields a bound on the number of incidences between m points and n planes in P 3 , with m ≥ n as O m √ n + mk , where k is the maximum number of collinear planes, provided that n = O(p 2 ) if p > 0. Examples show that this bound cannot be improved without additional assumptions.This gives one a vehicle to establish geometric incidence estimates when p > 0. For a non-collinear point set S ⊆ F 2 and a non-degenerate symmetric or skew-symmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is Ω min |S| 2 3 , p . This is also the best known bound over R, where it follows from the Szemerédi-Trotter theorem. Also, a set S ⊆ F 3 , not supported in a single semi-isotropic plane contains a point, from which Ω min |S| 1 2 , p distinct distances to other points of S are attained.

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Section: Resultssupporting

confidence: 60%

On the Number of Incidences Between Points and Planes in Three Dimensions

2017

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“…We start with a recent result of Shkredov & Vyugin [209]: [167] have given a series of bounds on #(G ± G), see also [173]. Many of this results are based on upper bounds for additive energy, that is 1 + a 2 = a 3 + a 4 : a 1 , a 2 , a 3 , a 4 ∈ G of G. Heath-Brown & Konyagin [120] have proved that for #G p 2/3 , we have…”

Section: Structured Setsmentioning

confidence: 97%

Additive Combinatorics over Finite Fields: New Results and Applications

Shparlinski¹

2013

Finite Fields and Their Applications

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“…Applying Proposition 14 with t 1 = t 2 , S(y) = y, T (y) = ay+b cy+d yields Corollary 15. For p large prime, t|(p − 1), t ≤ p 3 4 and U t = {y ∈ F p : y t = 1} the t-th roots of 1, Proof of Proposition 14: First we need a generalization of Proposition 3.2 in [VS12] where their common t is replaced by t 0 , t 1 , . .…”

Section: Hypothesismentioning

confidence: 99%